Midsets and Voronoi type decomposition with respect to closed convex sets

Abstract

Let Ωk denote the collection of all nonempty closed convex subsets of ℝk. We provide short proofs for the following: (i) {x ∈ ℝk: dist(x, A) = ϵ} is a C1-manifold of dimension k-1 for every A ∈ Ωk \ {ℝk} and ϵ > 0, (ii) {x ∈ ℝk: dist(x, A) = dist(x, B)} is a C1-manifold of dimension k - 1 for any two disjoint A, B Ωk.We also study the distance of points in ℝk to finitely many closed convex sets. Let k, n ≤ 2 and A = ∪j=1n Aj, where A1, ⋯, An ∈ Ωk are pairwise disjoint. We consider a Voronoi type decomposition of ℝk and establish some topological properties of its 'conflict set'. Letting Xp = {x ∈ ℝk: |{a ∈ A: ||x - a|| = dist(x,A)}| = p}, we prove with the help of result (ii) stated above that X1 ∪ X2 is a connected dense open subset of ℝk and that X2 = ∪p=2n Xp.